×

zbMATH — the first resource for mathematics

Decidable theories of pseudo-\(p\)-adic closed fields. (English. Russian original) Zbl 0717.12005
Algebra Logic 28, No. 6, 421-438 (1989); translation from Algebra Logika 28, No. 6, 643-669 (1989).
Let \(K\) a field and \(\Gamma\) an ordered Abelian group. A valuation \(\phi: K\to \Gamma \cup \{\infty \}\) is called \(p\)-valuation if \(\text{char}(K)=0\), \(\phi(p)\) is the least element of \(\Gamma\) greater than zero, and the residue class field is the \(p\)-element field. A field \(K\) is called P\(p\)C-field if \(\text{char}(K)=0\) and \(K\) is existentially closed in every regular totally \(p\)-adic extension of \(K\). We recall that the theory of the class of P\(p\)C-fields is undecidable. The author proves that the theory of maximal P\(p\)C-fields and the theories of the classes of P\(p\)C-fields with finitely generated absolute Galois groups are decidable.

MSC:
12L05 Decidability and field theory
03B25 Decidability of theories and sets of sentences
12J12 Formally \(p\)-adic fields
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] G. Cherlin, L. v. d. Dries and A. Macintyre, The Elementary Theory of Regularly Closed Fields, Preprint.
[2] Yu. L. Ershov, Decidability Problems and Constructive Models [in Russian], Nauka, Moscow (1980).
[3] Yu. L. Ershov, ”Regularly closed fields,” Dokl. Akad. Nauk SSSR,251, No. 4, 783–785 (1980). · Zbl 0467.03026
[4] Yu. L. Ershov, ”Totally real extensions of fields,” Dokl. Akad. Nauk SSSR,263, No. 5, 1047–1049 (1982).
[5] Ju. L. Ershov, ”Two theorems on regularly r-closed fields,” J. Reine Angew. Math.,347, 154–167 (1984). · Zbl 0514.12022
[6] Yu. L. Ershov, ”Realizable i-groups,” in: Some Problems and Tasks of Analysis and Algebra [in Russian], Novosibirsk State Univ., Novosibirsk (1985), pp. 46–60.
[7] Ju. L. Ershov, ”RRC-fields with small absolute Galois groups,” Ann. Pure Appl. Logic,43, 1989, 197–208 (1989). · Zbl 0691.12012
[8] M. D. Fried and M. Jarden, Field Arithmetic, Springer, Berlin (1986).
[9] C. Grob, ”Die Entscheidbarkeit der Theorie der maximalen pseudo p-adisch abgeschlossenen Körper,” Konstanzer Dissertationen, Band 202 (1987). · Zbl 0786.12006
[10] D. Haran and M. Jarden, ”The absolute Galois group of a pseudo p-adically closed field,” J. Reine Angew. math.,383, 147–206 (1988). · Zbl 0652.12010
[11] M. Jarden and U. Kiehne, ”The elementary theory of algebraic fields of finite corank,” Inv. Math.,30, 275–294 (1975). · Zbl 0315.12107
[12] A. Prestel, ”Pseudo real closed fields,” in: Set Theory and Model Theory, Proceedings, Bonn 1979 (Lect. Notes Math.,872), Springer, Berlin (1981), pp. 127–156.
[13] A. Prestel, ”Decidable theories of preordered fields,” Math. Annalen,258, 481–492 (1981/82). · Zbl 0478.12018
[14] A. Presetl and P. Roquette, Formally p-adic Fields (Lect. Notes Math.,1050), Springer, Berlin (1984).
[15] P. Ribenboim, Théorie des valuations, Les presses de l’Université de Montréal, Montréal (1965).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.